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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.transform;
import java.io.Serializable;
import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.complex.Complex;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.ArithmeticUtils;
import org.apache.commons.math3.util.FastMath;
/**
* Implements the Fast Cosine Transform for transformation of one-dimensional
* real data sets. For reference, see James S. Walker, <em>Fast Fourier
* Transforms</em>, chapter 3 (ISBN 0849371635).
* <p>
* There are several variants of the discrete cosine transform. The present
* implementation corresponds to DCT-I, with various normalization conventions,
* which are specified by the parameter {@link DctNormalization}.
* <p>
* DCT-I is equivalent to DFT of an <em>even extension</em> of the data series.
* More precisely, if x<sub>0</sub>, …, x<sub>N-1</sub> is the data set
* to be cosine transformed, the extended data set
* x<sub>0</sub><sup>#</sup>, …, x<sub>2N-3</sub><sup>#</sup>
* is defined as follows
* <ul>
* <li>x<sub>k</sub><sup>#</sup> = x<sub>k</sub> if 0 ≤ k < N,</li>
* <li>x<sub>k</sub><sup>#</sup> = x<sub>2N-2-k</sub>
* if N ≤ k < 2N - 2.</li>
* </ul>
* <p>
* Then, the standard DCT-I y<sub>0</sub>, …, y<sub>N-1</sub> of the real
* data set x<sub>0</sub>, …, x<sub>N-1</sub> is equal to <em>half</em>
* of the N first elements of the DFT of the extended data set
* x<sub>0</sub><sup>#</sup>, …, x<sub>2N-3</sub><sup>#</sup>
* <br/>
* y<sub>n</sub> = (1 / 2) ∑<sub>k=0</sub><sup>2N-3</sup>
* x<sub>k</sub><sup>#</sup> exp[-2πi nk / (2N - 2)]
* k = 0, …, N-1.
* <p>
* The present implementation of the discrete cosine transform as a fast cosine
* transform requires the length of the data set to be a power of two plus one
* (N = 2<sup>n</sup> + 1). Besides, it implicitly assumes
* that the sampled function is even.
*
* @since 1.2
*/
public class FastCosineTransformer implements RealTransformer, Serializable {
/** Serializable version identifier. */
static final long serialVersionUID = 20120212L;
/** The type of DCT to be performed. */
private final DctNormalization normalization;
/**
* Creates a new instance of this class, with various normalization
* conventions.
*
* @param normalization the type of normalization to be applied to the
* transformed data
*/
public FastCosineTransformer(final DctNormalization normalization) {
this.normalization = normalization;
}
/**
* {@inheritDoc}
*
* @throws MathIllegalArgumentException if the length of the data array is
* not a power of two plus one
*/
public double[] transform(final double[] f, final TransformType type)
throws MathIllegalArgumentException {
if (type == TransformType.FORWARD) {
if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
final double s = FastMath.sqrt(2.0 / (f.length - 1));
return TransformUtils.scaleArray(fct(f), s);
}
return fct(f);
}
final double s2 = 2.0 / (f.length - 1);
final double s1;
if (normalization == DctNormalization.ORTHOGONAL_DCT_I) {
s1 = FastMath.sqrt(s2);
} else {
s1 = s2;
}
return TransformUtils.scaleArray(fct(f), s1);
}
/**
* {@inheritDoc}
*
* @throws org.apache.commons.math3.exception.NonMonotonicSequenceException
* if the lower bound is greater than, or equal to the upper bound
* @throws org.apache.commons.math3.exception.NotStrictlyPositiveException
* if the number of sample points is negative
* @throws MathIllegalArgumentException if the number of sample points is
* not a power of two plus one
*/
public double[] transform(final UnivariateFunction f,
final double min, final double max, final int n,
final TransformType type) throws MathIllegalArgumentException {
final double[] data = FunctionUtils.sample(f, min, max, n);
return transform(data, type);
}
/**
* Perform the FCT algorithm (including inverse).
*
* @param f the real data array to be transformed
* @return the real transformed array
* @throws MathIllegalArgumentException if the length of the data array is
* not a power of two plus one
*/
protected double[] fct(double[] f)
throws MathIllegalArgumentException {
final double[] transformed = new double[f.length];
final int n = f.length - 1;
if (!ArithmeticUtils.isPowerOfTwo(n)) {
throw new MathIllegalArgumentException(
LocalizedFormats.NOT_POWER_OF_TWO_PLUS_ONE,
Integer.valueOf(f.length));
}
if (n == 1) { // trivial case
transformed[0] = 0.5 * (f[0] + f[1]);
transformed[1] = 0.5 * (f[0] - f[1]);
return transformed;
}
// construct a new array and perform FFT on it
final double[] x = new double[n];
x[0] = 0.5 * (f[0] + f[n]);
x[n >> 1] = f[n >> 1];
// temporary variable for transformed[1]
double t1 = 0.5 * (f[0] - f[n]);
for (int i = 1; i < (n >> 1); i++) {
final double a = 0.5 * (f[i] + f[n - i]);
final double b = FastMath.sin(i * FastMath.PI / n) * (f[i] - f[n - i]);
final double c = FastMath.cos(i * FastMath.PI / n) * (f[i] - f[n - i]);
x[i] = a - b;
x[n - i] = a + b;
t1 += c;
}
FastFourierTransformer transformer;
transformer = new FastFourierTransformer(DftNormalization.STANDARD);
Complex[] y = transformer.transform(x, TransformType.FORWARD);
// reconstruct the FCT result for the original array
transformed[0] = y[0].getReal();
transformed[1] = t1;
for (int i = 1; i < (n >> 1); i++) {
transformed[2 * i] = y[i].getReal();
transformed[2 * i + 1] = transformed[2 * i - 1] - y[i].getImaginary();
}
transformed[n] = y[n >> 1].getReal();
return transformed;
}
}